Finite Group Databasebeta

← #10399

Subgroups — #10399

Order 256.

347 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2626C211yesmaximal, characteristic
2#1872C211yesmaximal, characteristic, complemented
3#1852C211yesmaximal, characteristic, complemented
4C2×#328C211yesmaximal, characteristic, complemented
5C2×#323C211yesmaximal, characteristic
6#1439C211yesmaximal, characteristic
7#1294C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8C2211yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10C22×C22⋊C4—21no—
11C22×C22⋊C4C2211yescharacteristic
12C2×#107—21no—
13C2×#103—21no—
14C2×C22⋊C8C2211yescharacteristic
15C2×C22⋊C8C2211yescharacteristic
16#399—21no—
17#396—21no—
18#396—21no—
19#393—21no—
20#385—21no—
21#380C2211yescharacteristic
22#379—21no—
23C2×#95—21no—
24C2×#95C2211yescharacteristic
25#340—21no—
26#340—21no—
27#328—22no—
28#323—22no—
29C25D411yescharacteristic
30C22×Q8D411yescharacteristic
31C23×C4—21no—
32C23×C4—21no—
33C23×C4D411yescharacteristic
34C23×C4C2311yescharacteristic
35C22×C8—21no—
36C22×C8—21no—
37C22⋊Q8—21no—
38C22⋊Q8—21no—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4D411yescharacteristic
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D411yescharacteristic
50C2×C22⋊C4—21no—
51C2×C22⋊C4—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—42no—
63C2×C22⋊C4—42no—
64C2×C22⋊C4—42no—
65#107—42no—
66#103—42no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C22⋊C8D412yes—
71#95—21no—
72#95—21no—
73#95—21no—
74#95—21no—
75#95—21no—
76#95—42no—
77C24—21no—
78C24—21no—
79C24—21no—
80C24—21no—
81C24—21no—
82C24—21no—
83C24—41no—
84C24—41no—
85C24—41no—
86C24—41no—
87C24Q1612yes—
88C24C2×D411yescharacteristic
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—41no—
92C2×Q8—41no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4#4111yescharacteristic
122C22×C4#4111yescharacteristic
123C22×C4#4111yescharacteristic
124C2×C8—22no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—81no—
141C4⋊C4—81no—
142C22⋊C4—48no—
143C22⋊C4—41no—
144C22⋊C4—41no—
145C22⋊C4—41no—
146C22⋊C4—41no—
147C22⋊C4—41no—
148C22⋊C4—41no—
149C22⋊C4—41no—
150C22⋊C4—41no—
151C22⋊C4—81no—
152C22⋊C4—81no—
153C22⋊C4—81no—
154C22⋊C4—81no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—81no—
177C23—81no—
178C23—81no—
179C23—81no—
180C23—81no—
181C23—81no—
182C23#12011yescharacteristic
183C23#12111yescharacteristic
184C23#12311yescharacteristic
185C23C2×Q1611yescharacteristic
186C23#13511yescharacteristic
187C23AGL(1,ℤ/8)11yescharacteristic
188C23#13711yescharacteristic
189Q8—42no—
190Q8—81no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4#12111yescharacteristic
231C2×C4#12111yescharacteristic
232C2×C4#12311yescharacteristic
233C2×C4#12411yescharacteristic
234C8—42no—
235C8—81no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—82no—
253C22—82no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22#39411yescharacteristic
260C22#44911yescharacteristic
261C22#45111yescharacteristic
262C22#45311yescharacteristic
263C22#45511yescharacteristic
264C22B(4,𝔽2)11yescharacteristic
265C22#45811yescharacteristic
266C22#46711yescharacteristic
267C22#46811yescharacteristic
268C22#48411yescharacteristic
269C22#48511yescharacteristic
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—82no—
275C4—81no—
276C4—82no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4—81no—
281C2—21no—
282C2—21no—
283C2—21no—
284C2—21no—
285C2—82no—
286C2#160011yescharacteristic
287C2#160211yescharacteristic
288C2#161211yescharacteristic
289C2#161411yescharacteristic
290C2#203611yescharacteristic
291C2#203711yescharacteristic
292C2#204411yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1872C2semidirect
3. #1852C2semidirect
4. C2×#328C2semidirect
7. #1294C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#26262
#18722
#18522
C2×#3282
C2×#3232
#14392
#12942